几类带趋化项数学模型的动力学性质分析

发布时间：2018-03-27 10:21

本文选题：趋化性　切入点：食饵趋化　出处：《哈尔滨工业大学》2017年博士论文

【摘要】：趋化性是物种对环境中的化学信号进行检测以及响应然后作出的化学敏感运动的现象。依据生物对化学信号的靠近还是远离,可将趋化作用分为吸引的或者是排斥的。生物聚集是趋化现象的一个显著特征。在空间捕食的活动中,除了捕食者和食饵的随机运动外,同时还存在着趋化现象,即捕食者种群密度的时空变化也受食饵种群梯度的影响。因此,研究带食饵趋化项的捕食者-食饵模型以及分析细胞中的趋化模型都是很有必要的。本文主要研究了几类带趋化项模型的动力学性质,给出了这几类模型解的全局存在性及有界性。同时,还对捕食者-食饵系统做了进一步的研究,得到了系统的全局吸引子和系统的一致持久性。主要的工作包括以下几个方面:1.研究了光滑有界的区域中在无对流边界条件下带有扩散项和食饵趋化项的一般的捕食者-食饵模型解的全局存在性和有界性。这个结果对任意维空间区域中带有充分小的食饵趋化敏感系数的捕食者-食饵系统都是成立的。此外,给出了全局吸引子的存在性和系统的一致持久性的充分条件。特别地,把这些结果应用到了具体的三个模型中:第一,详细论述了带有扩散项和食饵趋化项的Rosenzweig-MacArthur捕食者-食饵模型并且应用以上结果,能够得到模型解的全局存在性和有界性以及全局吸引子的存在性和系统的一致持久性;第二,讨论了带有扩散项和食饵趋化项的强Allee效应的捕食者-食饵模型并且应用以上结果,由于(O,O)总是局部渐近稳定的,所以系统不是一致持久的,因此只能得到模型解的全局存在性和有界性以及全局吸引子的存在性;第三,考虑了带有趋化项的种群模型,并且利用以上结果进而推广了已有的一维模型的结果。2.研究了在齐次Neumann边界条件下的光滑有界区域上带有非线性趋化敏感函数项和增长项的吸引-排斥模型。利用能量估计的方法,证明了系统存在全局有界解并且给出了关于非线性趋化敏感函数项和增长项的充分条件。这个结果与带有线性项模型已有的结果相一致并且可以应用到更广泛的模型中。3.研究了在齐次Neumann边界条件下的光滑有界的凸区域上的吸引-排斥趋化模型。在这个模型中,当放缩比例常数为0且趋化敏感函数是非线性时,通过构造辅助函数的方法,给出了系统有唯一的全局一致有界古典解的充分条件;当放缩比例常数为1且趋化敏感函数X(v= χ0,ξ(w)=ξ0/w,其中χ0,ξ0是正常数,通过能量估计的方法,也得到了系统唯一的全局一致有界古典解的充分条件。
[Abstract]:Chemotaxis is a phenomenon in which species detect and respond to chemical signals in the environment and then make chemically sensitive movements. Chemotaxis can be classified as attractive or repellent. Biological aggregation is a prominent feature of chemotaxis. In space predation, in addition to the random movement of predators and prey, there is also chemotaxis. That is, the spatial and temporal variation of predator population density is also affected by the prey population gradient. It is necessary to study predator-prey models with prey chemotaxis and to analyze chemotaxis models in cells. The global existence and boundedness of the solutions of these models are given. At the same time, the predator-prey system is further studied. The global attractor of the system and the uniform persistence of the system are obtained. The main work includes the following aspects: 1. The general properties of a smooth bounded region with diffusion and prey chemotaxis under the condition of no convection boundary are studied. The global existence and boundedness of the solution of predator-prey model are all valid for predator-prey systems with sufficiently small prey chemotaxis sensitivity coefficients in arbitrary dimensional space. Sufficient conditions for the existence of global attractors and the uniform persistence of the system are given. In particular, these results are applied to three specific models: first, The Rosenzweig-MacArthur predator-prey model with diffusion term and prey chemotaxis term is discussed in detail. The global existence and boundedness of the model solution, the existence of global attractor and the uniform persistence of the system can be obtained by applying the above results. In this paper, the predator-prey model with strong Allee effect with diffusion term and prey chemotaxis term is discussed. By applying the above results, the system is not uniformly persistent because of its always locally asymptotically stable. Therefore, the global existence and boundedness of the solution and the existence of the global attractor can only be obtained. Thirdly, the population model with chemotaxis is considered. By using the above results, we generalize the results of the existing one-dimensional model. 2. We study the attractive repulsive model with nonlinear chemotaxis sensitive function term and growth term on the smooth bounded region under homogeneous Neumann boundary condition. Using the method of energy estimation, It is proved that there exists a global bounded solution for the system and a sufficient condition for the nonlinear chemotaxis sensitive function term and the growth term is given. This result is consistent with that of the model with linear term and can be applied to a wider range of. In the model .3. we study the attractive repulsive chemotaxis model on the smooth bounded convex region under homogeneous Neumann boundary conditions. When the scaling constant is 0 and the chemotactic sensitive function is nonlinear, a sufficient condition for the unique globally uniformly bounded classical solution of the system is given by constructing the auxiliary function. When the scaling constant is 1 and the chemotactic sensitivity function X ~ (v) = 蠂 ~ (0), 尉 _ (W) is a normal number, the sufficient conditions for the unique globally uniformly bounded classical solution of the system are also obtained by the method of energy estimation.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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